### Abstract

Motivated by recent ideas of Harman (Unif. Distrib. Theory, 2010) we develop a new concept of variation of multivariate functions on a compact Hausdorff space with respect to a collection D of subsets. We prove a general version of the Koksma-Hlawka theorem that holds for this notion of variation and discrepancy with respect to D. As special cases, we obtain Koksma-Hlawka inequalities for classical notions, such as extreme or isotropic discrepancy. For extreme discrepancy, our result coincides with the usual Koksma-Hlawka theorem. We show that the space of functions of bounded D-variation contains important discontinuous functions and is closed under natural algebraic operations. Finally, we illustrate the results on concrete integration problems from integral geometry and stereology.

Original language | English |
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Pages (from-to) | 773-797 |

Number of pages | 25 |

Journal | Journal of Complexity |

Volume | 31 |

Issue number | 6 |

Early online date | 16 Jun 2015 |

DOIs | |

Publication status | Published - 01 Dec 2015 |

### Keywords

- Hardy-Krause variation
- Harman variation
- Integral geometry
- Koksma-Hlawka theorem

### ASJC Scopus subject areas

- Algebra and Number Theory
- Statistics and Probability
- Numerical Analysis
- Control and Optimization
- Applied Mathematics

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## Cite this

*Journal of Complexity*,

*31*(6), 773-797. https://doi.org/10.1016/j.jco.2015.06.002